Abstract
We study the existence of pure Nash equilibria in congestion games with player-specific costs. Specifically, we provide a thorough characterization of the maximal sets of cost functions that guarantee the existence of a pure Nash equilibrium.
For the case that the players are unweighted, we show that it is necessary and sufficient that for every resource and for every pair of players the corresponding cost functions are affine transformations of each other. For weighted players, we show that in addition one needs to require that all cost functions are affine or all cost functions are exponential.
Finally, we construct a four-player singleton weighted congestion game where the cost functions are identical among the resources and differ only by an additive constant among the players and show that it does not have a pure Nash equilibrium. This answers an open question by Mavronicolas et al. [15] who showed that such games with at most three players always have a pure Nash equilibrium.
Supported by EPSRC grant EP/J019399/1 and the German Research Foundation (DFG) under contract KL 2761/1-1.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ackermann, H., Röglin, H., Vöcking, B.: Pure Nash equilibria in player-specific and weighted congestion games. In: Spirakis, P., Mavronicolas, M., Kontogiannis, S. (eds.) WINE 2006. LNCS, vol. 4286, pp. 50–61. Springer, Heidelberg (2006)
Ackermann, H., Skopalik, A.: On the complexity of pure Nash equilibria in player-specific network congestion games. In: Deng, X., Graham, F.C. (eds.) WINE 2007. LNCS, vol. 4858, pp. 419–430. Springer, Heidelberg (2007)
Anshelevich, E., Dasgupta, A., Kleinberg, J., Tardos, É., Wexler, T., Roughgarden, T.: The price of stability for network design with fair cost allocation. SIAM J. Comput. 38(4), 1602–1623 (2008)
Aumann, R.: Acceptable points in general cooperative n-person games. In: Luce, R.D., Tucker, A.W. (eds.) Contributions to the Theory of Games IV, pp. 287–324. Princeton University Press, Princeton (1959)
Dunkel, J., Schulz, A.: On the complexity of pure-strategy Nash equilibria in congestion and local-effect games. Math. Oper. Res. 33(4), 851–868 (2008)
Fotakis, D., Kontogiannis, S., Spirakis, P.: Selfish unsplittable flows. Theoret. Comput. Sci. 348(2-3), 226–239 (2005)
Gairing, M., Monien, B., Tiemann, K.: Routing (un-)splittable flow in games with player-specific linear latency functions. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. Part I. LNCS, vol. 4051, pp. 501–512. Springer, Heidelberg (2006)
Georgiou, C., Pavlides, T., Philippou, A.: Selfish routing in the presence of network uncertainty. Parallel Process. Lett. 19(1), 141–157 (2009)
Goemans, M., Mirrokni, V., Vetta, A.: Sink equilibria and convergence. In: Proc. 46th Annual IEEE Sympos. Foundations Comput. Sci., pp. 142–154 (2005)
Harks, T., Klimm, M.: On the existence of pure Nash equilibria in weighted congestion games. Math. Oper. Res. 37(3), 419–436 (2012)
Harks, T., Klimm, M., Möhring, R.: Characterizing the existence of potential functions in weighted congestion games. Theory Comput. Syst. 49(1), 46–70 (2011)
Konishi, H., Le Breton, M., Weber, S.: Equilibria in a model with partial rivalry. J. Econom. Theory 72(1), 225–237 (1997)
Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 404–413. Springer, Heidelberg (1999)
Libman, L., Orda, A.: Atomic resource sharing in noncooperative networks. Telecommun. Syst. 17(4), 385–409 (2001)
Mavronicolas, M., Milchtaich, I., Monien, B., Tiemann, K.: Congestion games with player-specific constants. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 633–644. Springer, Heidelberg (2007)
Milchtaich, I.: Congestion games with player-specific payoff functions. Games Econom. Behav. 13(1), 111–124 (1996)
Milchtaich, I.: The equilibrium existence problem in finite network congestion games. In: Spirakis, P., Mavronicolas, M., Kontogiannis, S. (eds.) WINE 2006. LNCS, vol. 4286, pp. 87–98. Springer, Heidelberg (2006)
Milchtaich, I.: Representation of finite games as network congestion games. In: Proc. 5th Internat. Conf. on Network Games, Control and Optimization, pp. 1–5 (2011)
Panagopoulou, P., Spirakis, P.: Algorithms for pure Nash equilibria in weighted congestion games. ACM J. Exp. Algorithmics 11, 1–19 (2006)
Rosenthal, R.: A class of games possessing pure-strategy Nash equilibria. Internat. J. Game Theory 2(1), 65–67 (1973)
Rosenthal, R.: The network equilibrium problem in integers. Networks 3, 53–59 (1973)
Tran-Thanh, L., Polukarov, M., Chapman, A., Rogers, A., Jennings, N.R.: On the existence of pure strategy Nash equilibria in integer-splittable weighted congestion games. In: Persiano, G. (ed.) SAGT 2011. LNCS, vol. 6982, pp. 236–253. Springer, Heidelberg (2011)
Voorneveld, M., Borm, P., van Megen, F., Tijs, S., Facchini, G.: Congestion games and potentials reconsidered. Int. Game Theory Rev. 1(3-4), 283–299 (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gairing, M., Klimm, M. (2013). Congestion Games with Player-Specific Costs Revisited. In: Vöcking, B. (eds) Algorithmic Game Theory. SAGT 2013. Lecture Notes in Computer Science, vol 8146. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41392-6_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-41392-6_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-41391-9
Online ISBN: 978-3-642-41392-6
eBook Packages: Computer ScienceComputer Science (R0)